! jupyter nbconvert --execute --to html "II.1. Single Factor Models.ipynb" --HTMLExporter.theme=dark
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=======
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generate default config file
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Continue notebook execution even if one of the cells throws an error and include the error message in the cell output (the default behaviour is to abort conversion). This flag is only relevant if '--execute' was specified, too.
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Write notebook output to stdout instead of files.
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Run nbconvert in place, overwriting the existing notebook (only
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Disable chromium security sandbox when converting to PDF..
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Shows code input. This flag is only useful for dejavu users.
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Overwrite base name use for output files.
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Examples
--------
The simplest way to use nbconvert is
> jupyter nbconvert mynotebook.ipynb --to html
Options include ['asciidoc', 'custom', 'html', 'html_ch', 'html_embed', 'html_toc', 'latex', 'markdown', 'notebook', 'pdf', 'python', 'qtpdf', 'qtpng', 'rst', 'script', 'selectLanguage', 'slides', 'webpdf'].
> jupyter nbconvert --to latex mynotebook.ipynb
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> jupyter nbconvert --to html --template lab mynotebook.ipynb
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c.NbConvertApp.notebooks = ["my_notebook.ipynb"]
> jupyter nbconvert --config mycfg.py
To see all available configurables, use `--help-all`.
[NbConvertApp] WARNING | pattern 'II.1. Single Factor Models.ipynb' matched no files
import yfinance as yf
from scipy import stats
import pandas as pd
import numpy as np
import plotly.express as px
from plotly.subplots import make_subplots
import plotly.graph_objects as go
import plotly.io as pio
######## FIX RELATIVE PATH
import sys
sys.path.append("../../")
import plotly_template
pio.templates.default = 'simple_white+blog_mra'
def to_jupyter_latex(df):
print(df.head().to_latex(index=True, float_format="{:.2f}".format,).replace('\\toprule', '\\hline').replace('\\midrule', '\\hline').replace('\\bottomrule','\\hline').replace('tabular', 'array'))
The CAPM equation suggests that the expected return on an asset is equal to the risk-free rate plus a risk premium. The risk premium is determined by the asset's beta, which measures its sensitivity to systematic risk. Systematic risk refers to the risk that cannot be eliminated through diversification and is associated with the overall market movements.
\begin{equation} E(R_{i}) = R_{f} + \beta_{i}(E(R_{M}) - R_{f}) \end{equation}Reference to equation $\eqref{eq:vector_ray}$
df = pd.read_excel(r"data/Examples_II.1.xls", sheet_name="EX_II.1.1").set_index("Date")
df.index = df.index.date
df = df.loc[:, :"MSFT"]
ret = df.pct_change()[1:] #returns
reg_msft = stats.linregress(ret.SPX, ret.MSFT) # regress MSFT on SPX
reg_nwl = stats.linregress(ret.SPX, ret.NWL) # regress NWL on SPX
reg_msft
LinregressResult(slope=1.104210318384485, intercept=-0.0006618592810449706, rvalue=0.5754570214718748, pvalue=1.8014269223103655e-36, stderr=0.07885975545252356, intercept_stderr=0.0017892738852325724)
### application of formula above
b = sum((ret["MSFT"] - ret["MSFT"].mean())*(ret["SPX"] - ret["SPX"].mean())) / sum((ret["SPX"] - ret["SPX"].mean())**2)
print(b)
a = ret["MSFT"].mean() - b*ret["SPX"].mean()
print(a)
1.1042103183844845 -0.0006618592810449704
beta_msft = reg_msft.slope
alpha_msft = reg_msft.intercept
stderr_msft = reg_msft.stderr
annual_vol_msft = stderr_msft * np.sqrt(52)
beta_nwl = reg_nwl.slope
alpha_nwl = reg_nwl.intercept
stderr_nwl = reg_nwl.stderr
annual_vol_nwl = stderr_nwl * np.sqrt(52)
print("MSFT")
print("Alpha: "+str(round(alpha_msft, 4)))
print("Beta: "+str(round(beta_msft, 4)))
print("Std Err: "+str(round(alpha_msft, 4)))
print("Annual Vol: "+str(round(annual_vol_msft, 4)))
print("")
print("WMT")
print("Alpha: "+str(round(alpha_nwl, 4)))
print("Beta: "+str(round(beta_nwl, 4)))
print("Std Err: "+str(round(stderr_nwl, 4)))
print("Annual Vol: "+str(round(annual_vol_nwl, 4)))
MSFT Alpha: -0.0007 Beta: 1.1042 Std Err: -0.0007 Annual Vol: 0.5687 WMT Alpha: 0.0036 Beta: 0.506 Std Err: 0.071 Annual Vol: 0.5118
Assume to construct a portfolio which invests 70% in MSFT and 30% in WMT
At stocks level
print(alpha_msft*0.3 + alpha_nwl*0.7)
print(beta_msft*0.3 + beta_nwl*0.7)
0.0023084375088988653 0.6854319125642749
At portfolio level
#define weights
w = np.full((len(ret), 2), [0.3, 0.7])
# portfolio returns
ret["PTF"] = ret[["MSFT", "NWL"]].mul(w).sum(axis=1)
reg_ptf = stats.linregress(ret.SPX, ret.PTF)
alpha_ptf = reg_ptf.intercept
beta_ptf = reg_ptf.slope
print("Ptf alpha: " +str(reg_ptf.intercept))
print("Ptf beta: " +str(reg_ptf.slope))
Ptf alpha: 0.0023084375088988653 Ptf beta: 0.6854319125642752
linspace = np.arange(-0.1, 0.1, 0.01)
regline_msft = alpha_msft + beta_msft*linspace
regline_nwl = alpha_nwl + beta_nwl*linspace
regline_ptf = alpha_ptf + beta_ptf*linspace
fig = make_subplots(rows=1, cols=3, shared_yaxes='all', subplot_titles=("MSFT", "NWL", "PTF"))
fig.add_trace(
go.Scatter(x=ret.SPX, y=ret.MSFT, mode="markers"),
row=1, col=1)
fig.add_trace(go.Scatter(x=linspace, y=regline_msft), row=1, col=1)
fig.add_trace(
go.Scatter(x=ret.SPX, y=ret.NWL, mode="markers"),
row=1, col=2)
fig.add_trace(go.Scatter(x=linspace, y=regline_nwl), row=1, col=2)
fig.add_trace(
go.Scatter(x=ret.SPX, y=ret.PTF, mode="markers"),
row=1, col=3)
fig.add_trace(go.Scatter(x=linspace, y=regline_msft), row=1, col=3)
fig.add_trace(go.Scatter(x=linspace, y=regline_nwl), row=1, col=3)
fig.add_trace(go.Scatter(x=linspace, y=regline_ptf), row=1, col=3)
fig.update_layout(height=600, width=900, title_text="Regression over SPX")
fig.show()
df = pd.read_excel(r"data/Examples_II.1.xls", sheet_name="EX_II.1.2").set_index("Date")
df = df.loc[:, :"Cisco"]
df.index = df.index.date
ret = np.log(df/df.shift())[1:] #returns
reg_amex = stats.linregress(ret.SP100, ret.Amex)
reg_cisco = stats.linregress(ret.SP100, ret.Cisco)
# standard error Amex
stderr_amex = (((ret.Amex - (reg_amex.intercept + reg_amex.slope*ret["SP100"]))**2).sum() / (len(ret)-2))**0.5
# standard error Cisco
stderr_cisco = (((ret.Cisco - (reg_cisco.intercept + reg_cisco.slope*ret["SP100"]))**2).sum() / (len(ret)-2))**0.5
At stock level (assuming uncorrelated returns)
print("s = "+str(round (((stderr_amex**2)*(0.6**2) + (stderr_cisco**2)*(0.4**2))**0.5 *np.sqrt(250), 4)))
s = 0.1998
At portfolio level
w = np.full((len(ret), 2), [0.6, 0.4])
# portfolio returns
ret["PTF"] = ret[["Amex", "Cisco"]].mul(w).sum(axis=1)
reg_ptf = stats.linregress(ret.SP100, ret.PTF)
# standard error Cisco
stderr_ptf = (((ret.PTF - (reg_ptf.intercept + reg_ptf.slope*ret["SP100"]))**2).sum() / (len(ret)-2))**0.5
print("s= " +str(round(stderr_ptf*np.sqrt(250), 4)))
s= 0.1819
Note: For returns, applying weights to alpha and betas for each stocks yields the same results as by using an OLS regression on the portfolio returns. However, it does make a difference for the specific risk of the portfolio!
Why: The reason is that specific risks are not uncorrelated
Solution: To compute specific risk correcly, it is recommended to first construct the constant weighted portfolio and then to apply OLS regression. Alternatively, we can calculate the covariance matrix with the regression residuals.
w_amex=0.6
lam=0.95
sq_ret = ret**2
df_ewma_var = pd.DataFrame(columns=sq_ret.columns, index=sq_ret.index)
ewma = np.array([0.0001, 0.0001, 0.0001, 0.0001])
for row, item in sq_ret.iterrows():
ewma = (1-lam)*item + ewma*lam
df_ewma_var.loc[row, :] = ewma.values
mul_ret = ret[["Amex", "Cisco", "PTF"]].mul(ret["SP100"], axis=0)
df_ewma_cov = pd.DataFrame(columns=["Amex", "Cisco", "PTF"], index=sq_ret.index)
ewma = np.array([0.0001, 0.0001, 0.0001])
for row, item in mul_ret.iterrows():
ewma = (1-lam)*item + ewma*lam
df_ewma_cov.loc[row, :] = ewma.values
emwa_beta = df_ewma_cov.div(df_ewma_var["SP100"], axis=0).iloc[:-1]
emwa_relvol = (df_ewma_var[["Amex", "Cisco", "PTF"]].div(df_ewma_var["SP100"], axis=0))**0.5
emwa_corr = emwa_beta/emwa_relvol
sys_risk = emwa_beta["PTF"]* (df_ewma_var["SP100"]*250)**0.5
emwa_beta["Ols_beta"] = reg_ptf.slope
fig = make_subplots(specs=[[{"secondary_y": True}]])
fig.add_trace(go.Scatter(x=emwa_beta.index, y=emwa_beta.PTF, name="EWMA Beta"), )
fig.add_trace(go.Scatter(x=sys_risk.index, y=sys_risk.values, name="Systematic Risk"), secondary_y=True)
fig.add_trace(go.Scatter(x=emwa_beta.index, y=emwa_beta.Ols_beta, line_color="black", name="OLS Beta"))
fig.update_layout(height=600, width=900, title_text="Portfolio EWMA vs OLS Beta")
fig.show()
fig = make_subplots(specs=[[{"secondary_y": True}]])
fig.add_trace(go.Scatter(x=emwa_beta.index, y=emwa_beta.Amex, name="EWMA Beta Amex "), )
fig.add_trace(go.Scatter(x=emwa_relvol.index, y=emwa_relvol.Amex, name= "EWMA RelVol Amex"))
fig.add_trace(go.Scatter(x=emwa_corr.index, y=emwa_corr.Amex, name= "EWMA Corr Amex"), secondary_y=True)
fig.update_layout(yaxis = dict(range=[0, 5]), yaxis2 = dict(range=[0, 1]), title_text="Amex")
fig.show()
fig = make_subplots(specs=[[{"secondary_y": True}]])
fig.add_trace(go.Scatter(x=emwa_beta.index, y=emwa_beta.Cisco, name="EWMA Beta Cisco"))
fig.add_trace(go.Scatter(x=emwa_relvol.index, y=emwa_relvol.Cisco, name="EWMA RelVol Cisco"))
fig.add_trace(go.Scatter(x=emwa_corr.index, y=emwa_corr.Cisco, name="EWMA Corr Cisco"), secondary_y=True)
fig.update_layout(yaxis = dict(range=[0, 5]), yaxis2 = dict(range=[0, 1]), title_text="Cisco")
fig.show()
Taking the expectation and variance of (reference) and assuming $Cov(X, \epsilon) = 0$, we can see that: $$ E(Y) = \alpha + \beta E(X) $$ $$ V(Y) = \beta^2V(X) + V(\epsilon)$$ Hence the portfolio volatility can be decomposed into:
The (reference) can then be expressed as $$ TotalRisk = (SystemicRisk^2 + SpecificRisk^2)^{1/2} $$
Note:
TODO: